Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

Oberdieck, Georg

doi:10.2140/gt.2018.22.323

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Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

Georg Oberdieck

Geometry & Topology 22 (2018) 323–437

DOI: 10.2140/gt.2018.22.323

Abstract

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.

Let $S$ be a K3 surface and let ${Hilb}^{d} (S)$ be the Hilbert scheme of $d$ points of $S$ . In the case of elliptically fibered K3 surfaces $S \to ℙ^{1}$ , we calculate genus-0 Gromov–Witten invariants of ${Hilb}^{d} (S)$ , which count rational curves incident to two generic fibers of the induced Lagrangian fibration ${Hilb}^{d} (S) \to ℙ^{d}$ . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.

We also prove results for genus-0 Gromov–Witten invariants of ${Hilb}^{d} (S)$ for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov–Witten invariants of the Hilbert scheme of two points of $ℙ^{1} \times E$ , where $E$ is an elliptic curve.

Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on ${Hilb}^{d} (S)$ with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of $S$ . We prove the conjecture in the first nontrivial case ${Hilb}^{2} (S)$ . As a corollary, we find that the full genus-0 Gromov–Witten theory of ${Hilb}^{2} (S)$ in primitive classes is governed by Jacobi forms.

We present two applications. A conjecture relating genus-1 invariants of ${Hilb}^{d} (S)$ to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when $d = 2$ . Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.

Keywords

Gromov–Witten invariants, K3 surfaces

Mathematical Subject Classification 2010

Primary: 14N35

Secondary: 14J28, 11F50

References

Publication

Received: 11 November 2015

Revised: 1 March 2017

Accepted: 30 March 2017

Published: 31 October 2017

Proposed: Jim Bryan

Seconded: Richard Thomas, Dan Abramovich

Authors

Georg Oberdieck
Department Mathematik ETH Zürich Zürich Switzerland	Department of Mathematics MIT Cambridge, MA United States

Volume 22, issue 1 (2018)